Two Equations, Two Unknowns

April 16th, 2017

S: Before we begin, I have two things to say to whoever is reading this.  1: I’m really sick and I’m losing my voice, so can’t talk, so I’m literally just talking on paper and my dad has to read what I write to know what I’m saying.  2: My dad was being really crabby about the Beatle number system, so, unfortunately, I have decided to stop using it.

J: Problem #1:

S: Problem #John!!!  Just kidding!  I promise I won’t do that again.

J: Problem #1: x is the number of songs that John wrote this month, and y is the number of songs that Paul wrote this month.

S: Which month?  If you’re talking about April 2017, keep in mind that it’s impossible for x to be anything other than zero.

J: The problem does not specify which month.

S: You’re making the problem up, though!

J: Are we ready?

S: 😡

S: Fine.

J: I can go to the other room and Google up some actual data.

S: Or you could make up a month.

J: Beatleary.

S: U NO WOT I MEANT!!!  Fine.  Just tell me the problem.

J: Well, the fan club newsletter-

S: -run by Freda Kelly-

J: -gives a report of two equations, two unknowns and asks the question, “Who is ahead for this month and by how much?” and…

x + 2y = 13

3x – y = 4

S: Those are really simple compared to what’s on School Yourself (School Yourself is where I do math, and it’s a wonderful learning resource!  Link: www.schoolyourself.org).  But I’ll do the problem.

x = 13 – 2y

3 · (13 – 2y) – y = 4

3 · 13 – 3 · 2y – y = 4

39 – 6y – y = 4

-6y – y = 4 – 39

-7y = -35

y = -35/-7

y = 5

Time to plug it into the other equation…

x = 13-10

x = 3

x = 3 and y = 5.

You did that on purpose because Paul is my favorite, didn’t you?

J: Yes.

S: Knew it.  Next problem!

J: It’s a different month.  Same problem.

S: WHAT???  Be creative!  We have an audience!

J: Beatlember.

S:

J: Is that not creative?

S:

J: Oh.  Our AUDIENCE.

J: George and Ringo met some girls.  The number of girls George met is x, and the number of girls Ringo met is y.  How many did they meet at the party in question all together?

S: Now.

J:

J:

J: x² + 2y = 6

3x + 2y² = 8

S: Somewhere here, I’ll have to use the quadratic formula.  Which I forgot.  Luckily, it’s written down.  Oh, and by the way, for those of you who don’t know the quadratic formula, it’s (-B ± √B² – 4AC) ÷ 2A.

J: You don’t actually have to use it.

S: Oh.  But you said I did!

J: I recommend just doing regular substitution, as you did in the last problem.

S: Whatever “regular substitution” means.

J: Get x all alone.  And then substitute x‘s value into the second equation and solve for y.

S: Okay.  I’ll write out the equation again and try to solve it…

x² + 2y = 6

3x + 2y² = 8

2y = 6 – x²

y = 6/2 – x²/2

y = 3 – x²/2

3x + 2 (3 – x²/2)² = 8

WOW.

3x + 6 – 2 · (x²/2)² = 8

6 – 2 · (x²/2)² = 8 – 3x

How do I get x out of the parentheses?

J: So, for taking the term in the parentheses to a power, we could simply use FOIL-

S: I hate FOIL.  Is that the only way?

J: The other way is to square the numerator and square the denominator.

S: I’ll do that.

6 – 2 · x^{2^2}/2² = 8 – 3x

J: I don’t think you’re solving it correctly.  When you distributed the 2, you forgot to keep the ² with the 3.

S: Okay, I’ll go back…  Once again, I have:

3x + 2 (3 – x²/2)² = 8

3x + 18 + 2 (x²/2)² = 8

J: That is not the same equation.  FOIL is the reality.  Multiply the (3 – x²/2) by itself.  Then multiply all of those terms by the 2.

S: Okay.  3x + (3 – x²/2) · (3 – x²/2) · 2 = 8

J: Correct.  Now we can FOIL the two terms in parentheses.

S: Alright.  I hate FOIL, but I’ll try.

F     3 · 3

O     3 · – x²/2

I     – x²/2 · 3

L     – x²/2 · – x²/2

3 · 3 = 9, and everything else is the same.  So, 9 + 3 · – x²/2 + – x²/2 · 3 + – x²/2 · – x²/2

J: Yeah.  So that can be further simplified.

S: How?

J: Well, you can put the numbers first.

S: So, 9 + – 3 · x²/2 + – 3 · x²/2 + – x²/2 · – x²/2

And 9 + – 6 · x²/2 + – x²/2 · – x²/2 = 8

And 3x + 2 (9 + -6 · x²/2 + x²/2 · x²/2) = 8

We’re back in parentheses.

J: And we distribute the 2.

S: Which I tried to do in the first place.

J: But it didn’t get to the first term; it only went to the first and then it went sideways.  By the way, this is an unusually hard problem.  I don’t think you did problems like this on School Yourself.

S: The inequalities seemed this hard.

J: Inequalities are hard for a different reason.

S: So, distributing the 2…

3x + 18 + 2 (-6 · x²/2) + 2 (x²/2 · x²/2) = 8

We’re back in those dumb parentheses!  And the Beatles haven’t come up in paragraphs and paragraphs!

J: Well, let’s simplify-

S: BEATLES!!!

J: -the 2 (-6 · x²/2), which is equal to -6x².

S: 3x– GAAAAAAH I’M SO SICK OF THIS I WANT TO SLEEP THE BEATLES ARE PRACTICALLY ABSENT FROM THIS WHOLE THING!!!

J: John is very sick and is having trouble solving this problem, and he is very impatient with Paul’s attempts to help him.

S: Okay.  What can we do about it?  We aren’t anywhere near that time or place.

J: We can simplify the equation further and try to find out how many girls George meets and hope that they will somehow peer through a black hole and see the answer.

S: Alright.  3x + 18 + -6x² + 2 x⁴/4 = 8

Okay.  18 + -6x² + 2 x⁴/4 = 8 – 3x

-6x² +  2 x⁴/4 = -10 – 3x

-6x² + x⁴/2 = -10 – 3x

J: I’d leave them on the other side.

S: 😠

J: Okay, I think it’s time to just take some guesses on how many girls George met.  He can’t have met 0 or a negative number.

S: He could have met 0.

J: So, let’s plug in 0 and see if the equality is true.

S: NO.  Because you said it wasn’t, and I don’t want to waste time because I am sick and tired.

-6 · 0² + 0⁴/2 = -10 – 3 · 0

We can just see that 0 doesn’t work because anything multiplied by 0 is 0, and then everything would just be 0.

J: How about 1?

S: That changes nothing, though.  Well, I guess…

-6 · 1² + 1⁴/2 = -10 – 3 · 1

-6 + 0.5 = -13

-6 + 0.5 ≠ -13

1 is incorrect!  I don’t want to go on!

J: You must!  Try 2 next.

S: I caaaaaaaaaan’t!!!

J: Please please me and try 2.

S: Fine.

-6 · 2² + 2⁴/2 = -10 – 6

-24 + 8 = -16

FINALLY!!!  GLORY IZ MINE!!!

J: You know how many young women George met.

S: -16.  Wait a second!

J: The equality is true when you plugged in 2, so 2 satisfies the equation.

S: Okay.  So, back to the two original equations, with x as 2:

2² + 2y = 6

2y = 6 – 2²

2/2y = 2/2

y = 1

George met two girls and Ringo met one.  John and Paul peered into the black hole and found the answer!  Now John can go to bed.